Empirical Risk on Intervened Distribution. The empirical risk minimization for learning causal effect \(P(Y|do(X=x))\)  has been well formulated in [1]}. We define the empirical risk \(\mathcal {R}\)  of the estimator \(f\)  on the intervened distributions as:
\(\begin{aligned}\mathcal {R}(f)&=\mathbb {E}_{x\sim P(X),y\sim P(Y|do(X=x))}\mathcal {L}(y,f(x))\\&=\sum _x\sum _y\mathcal {L}(y,f(x))P(y|do(x))P(x),\end{aligned}\) 
